HES 505 Fall 2022: Session 16
Matt Williamson
Define a point process and their utility for ecological applications
Define first and second-order Complete Spatial Randomness
Use several common functions to explore point patterns
Leverage point patterns to interpolate missing data
Point pattern: A set of events within a study region (i.e., a window) generated by a random process
Set: A collection of mathematical events
Events: The existence of a point object of the type we are interested in at a particular location in the study region
A marked point pattern refers to a point pattern where the events have additional descriptors
Some notation:
\(S\): refers to the entire set
\(\mathbf{s_i}\) denotes the vector of data describing point \(s_i\) in set \(S\)
\(\#(S \in A )\) refers to the number of points in \(S\) within study area \(A\)
The pattern must be mapped on a plane to preserve distance
The study area, \(A\), should be objectively determined
There should be a \(1:1\) correspondence between objects in \(A\) and events in the pattern
Events must be proper i.e., refer to actual locations of the event
For some analyses the pattern should be a census of the relevant events
Modeling random processes means we are interested in probability densities of the points (first-order)
Also interested in how the presence of some events affects the probability of other events (second-order)
Finally interested in how the attributes of an event affect location (marked)
\[ \begin{equation} \hat{f}(x) = \frac{1}{nh_xh_y} \sum_{i=1}^n k\bigg(\frac{{x-x_i}}{h_x},\frac{{y-y_i}}{h_y} \bigg) \end{equation} \]
Assume each location in \(\mathbf{s_i}\) drawn from unknown distribution
Distribution has probability density \(f(\mathbf{x})\)
Estimate \(f(\mathbf{x})\) by averaging probability “bumps” around each location
Need different object types for most operations in R (as.ppp)
\(h\) is the bandwidth and \(k\) is the kernel
We can use stats::density to explore
Small values for \(h\) give ‘spiky’ densities
Large values for \(h\) smooth much more
Some kernels have optimal bandwidth detection
tmap package (later) provides additional functionality
KDEs assume independence of points (first order randomness)
Second-order methods allow dependence amongst points (second-order randomness)
Several functions for assessing second order dependence (\(K\), \(L\), and \(G\))
\[ \begin{equation} K(d) = \lambda^{-1}E(N_d) \end{equation} \]
\[ \begin{equation} K_{CSR}(d) = \pi d^2 \end{equation} \]
When working with a sample the distribution of \(K\) is unknown
Estimate with
\[ \begin{equation} \hat{K}(d) = \hat{\lambda}^{-1}\sum_{i=1}^n\sum_{j=1}^n\frac{I(d_{ij} <d)}{n(n-1)} \end{equation} \]
where:
\[ \begin{equation} \hat{\lambda} = \frac{n}{|A|} \end{equation} \]
spatstat packageGenerating 99 simulations of CSR ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99.
Done.
\(L\) function: square root transformation of \(K\)
\(G\) function: cumulative distribution of nearest neighbor distances
Goal: estimate the value of \(z\) at new points in \(\mathbf{x_i}\)
Most useful for continuous values
Nearest-neighbor, Inverse Distance Weighting, Kriging
\[ \begin{equation} \hat{z}(\mathbf(x)) = \frac{\sum_{i=1}w_iz_i}{\sum_{i=1}w_i} \end{equation} \]